# Thread: Limits / Series question 2

1. ## Limits / Series question 2

Let a1 = 1 and an = n(an-1 + 1 ) for n=2,3,.....

Define

P = (1+ 1/a1 )(1+1/a2)......(1+1/an )

The Limit P when n tends to infinity is?

2. Originally Posted by champrock
Let a1 = 1 and an = n(an-1 + 1 ) for n=2,3,.....

Define

P = (1+ 1/a1 )(1+1/a2)......(1+1/an )

The Limit P when n tends to infinity is?
Step 1. Work out the first few terms to get an idea of what is happening. You should find that the first few values of $\displaystyle a_n$ are 1, 4, 15, 64, 325, 1956, and that the corresponding value of P is $\displaystyle \tfrac21.\tfrac54.\tfrac{16}{15}.\tfrac{65}{64}.\t frac{326}{325} .\tfrac{1957}{1956}$. This simplifies to $\displaystyle \frac{1957}{6!} \approx2.718$, which looks suspiciously close to e.

Now see if you can turn those observations into proofs. You need to show first that the product $\displaystyle (1+ 1/a_1 )(1+1/a_2)\ldots(1+1/a_n )$ is equal to $\displaystyle \frac{a_n+1}{n!}$. Then you want to show that $\displaystyle \lim_{n\to\infty}\frac{a_n}{n!} = e$.

Hint for that last part: let $\displaystyle b_n = \frac{a_n}{n!}$. Show that $\displaystyle b_n = b_{n-1}+\frac1{(n-1)!}$, and deduce that $\displaystyle b_n = 1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\ldots+\frac1{( n-1)!}$.

3. answer is given as (e+1). ..